Modern Physics: Chapter 7 – Introduction to Wavefunctions

XKCD/849

So I already mentioned that everything has a matter-wave associated with it. Developing the idea a bit further, we get the concept of a wavefunction which, by convention, is denoted by $\psi$.

Each physical system is described by a state function which determines all can be known about the system. … The wavefunction has to be finite and single valued throughout position space, and furthermore, it must also be a continuous and continuously differentiable function.

Furthermore, a wavefunction is generally a complex function i.e it consists of both a real part and an imaginary part.

• $\psi (x, t)$ – A wavefunction

To get the probably of a particle being at a position $x$, you have to get a real value out of the complex wavefunction. Hence the probably of finding a particle described by a wavefunction $\psi$ is the product of the function $\psi$ at $x$ and its complex conjugate $\psi^{*}$ at $x$.

• $\psi (x_{0}) \psi^{*}(x_{0})$ – probability of finding the particle at $x_{0}$.

To get the probability from a range $x \in (x_{0}, x_{1})$ where $x$ is a continous variable, you can take the integral of $\psi^{*}\psi$ from $x_{0}$ to $x_{1}$:

• $P(x) = \int_{x_{0}}^{x_{1}} \psi^{*}(x) \psi(x) dx$